Equal power sets -> Equal sets?

Hey guys, quick question, and I apologize if it's already been asked recently. But is it possible to prove two sets are equal if they have the same power sets? That is, can you say that A=B if A and B have the same power set?

I have a feeling the answer's no, but I can't think of an example where it wouldn't be true.

Hey guys, quick question, and I apologize if it's already been asked recently. But is it possible to prove two sets are equal if they have the same power sets? That is, can you say that A=B if A and B have the same power set?

I have a feeling the answer's no, but I can't think of an example where it wouldn't be true.

Re: Equal power sets -> Equal sets?

If two sets, X and Y, have the same power sets (the power set of set A is the collection of all subsets of A) then the singleton sets in the power sets must be the same. But X is just equal to the union of all singleton sets in the power set of X and Y is the union of all singleton sets in the power set of Y.

Re: Equal power sets -> Equal sets?

Or seeing it slightly differently; since A and B have the same powersets (thus they have the same subsets) and every set is a subset of itself, $\displaystyle A \subseteq B$ and $\displaystyle B \subseteq A$, so $\displaystyle A = B$